Bi-Scalar Fishnet Theory

Updated 4 September 2025

Bi-scalar fishnet theory is a 4D, nonunitary conformal field theory derived from a gamma-deformed N=4 SYM double-scaling limit, characterized by its fishnet Feynman graphs.
Its planar perturbative expansion features regular square lattice diagrams constrained by Yangian symmetry and integrable spin chain structures.
The theory enables exact computation of multi-loop correlators and scattering amplitudes, serving as a benchmark for exploring quantum integrability in high-dimensional QFT.

The bi-scalar fishnet theory is a four-dimensional, nonunitary, integrable conformal field theory defined as a particular double-scaling limit of the gamma-deformed $\mathcal{N}=4$ super Yang-Mills theory. Its perturbative expansion in the planar limit is characterized by “fishnet” Feynman graphs—regular square lattices of scalar propagators—which are tightly constrained by integrability and exhibit an infinite-dimensional Yangian symmetry. This model has served as a laboratory for exploring the interplay between quantum integrability, conformal symmetry, and the computation of multi-loop correlation functions and scattering amplitudes.

1. Definition and Construction

The bi-scalar fishnet theory is constructed as follows:

Lagrangian: The action in four dimensions consists of two complex adjoint scalar fields $\phi_1$ and $\phi_2$ :

$\mathcal{L} = \frac{N_c}{2}\left(\partial^\mu \phi_1^\dagger \partial_\mu \phi_1 + \partial^\mu \phi_2^\dagger \partial_\mu \phi_2 + 2\xi^2 \phi_1^\dagger \phi_2^\dagger \phi_1 \phi_2\right)$

Here, $\xi$ is the effective coupling and $N_c$ the rank of the gauge group.

Double-Scaling Limit: The theory emerges from $\gamma$ -deformed $\mathcal{N}=4$ SYM in a double-scaling limit where the 't Hooft coupling $g\to0$ and the twist parameter $e^{-i\gamma/2}\to\infty$ , so that $\xi=g e^{-i\gamma/2}$ remains finite. This decouples most degrees of freedom, leaving only the bi-scalar sector with its chiral quartic interaction (Chicherin et al., 2017).
Planar Limit: At large $N_c$ , only a unique regular fishnet graph survives at each loop order in the perturbative expansion. The bulk of every planar Feynman diagram is given by a square lattice; non-planar corrections are suppressed (Chicherin et al., 2017, Gromov et al., 2017).

2. Fishnet Feynman Diagrams and Integrable Structures

Fishnet Graphs: In the bi-scalar theory, all perturbative amplitudes and correlators are represented by a single fishnet Feynman graph with the topology of a disc (for amplitudes) or a cylinder/wheel (for two-point correlators). Such a graph is a region cut from a regular square lattice of alternating $\phi_1$ and $\phi_2$ propagators (Chicherin et al., 2017, Kazakov, 2018).
Spin Chains and R-Matrix: The iterative construction of these graphs is governed by a graph-building operator that can be identified with the transfer matrix of an integrable non-compact $SU(2,2)$ (or $SO(1,5)$ ) Heisenberg spin chain. The underlying algebraic structures include conformal Lax operators and associated $R$ -matrices, whose factorization ensures integrability (Gromov et al., 2017, Kazakov, 2018).
Separation of Variables and Star-Triangle Relation: Computation of multi-loop integrals utilizes Sklyanin’s separation of variables method. The star–triangle relation (uniqueness) is a key identity allowing for the exact evaluation and transformation of lattice Feynman integrals, guaranteeing their integrability (Derkachov et al., 2021, Derkachov et al., 2022).

3. Yangian Symmetry and Its Realization

Yangian Algebra: The fishnet graphs are eigenfunctions of a Yangian $Y[\mathfrak{so}(2,4)]$ , which appears as an infinite-dimensional symmetry algebra extending the conformal algebra. The RTT realization packages these generators into a monodromy matrix $T(u)$ depending on a spectral parameter $u$ :

$T(u) = 1 + u^{-1} J^{(0)} + u^{-2} J^{(1)} + \ldots$

where $J^{(0)}$ are the ordinary conformal generators and $J^{(1)}$ the level-one Yangian generators (Chicherin et al., 2017).

Boundary Monodromy and Inhomogeneities: Yangian invariance is achieved by wrapping the disc boundary of the fishnet graph with a product of Lax operators with carefully assigned inhomogeneity parameters—incremented or decremented depending on the polygonal geometry’s turns. This ensures the external sites carry well-defined conformal weights and the monodromy acts diagonally, leading to stringent differential constraints on the integrals (Chicherin et al., 2017).
Dual Conformal Symmetry: There is a tight relationship between the level-one Yangian generators and dual conformal symmetry—in momentum space, the former reproduces the generator of dual conformal boosts. The invariance under dual conformal symmetries reflects the exceptional IR finiteness and analytic control of amplitudes in the model (Chicherin et al., 2017, Chicherin et al., 2022).

4. Exact Computations: Correlators, Amplitudes, and Spectrum

Four-Point Correlators and OPE Data: All-loop correlation functions, such as four-point functions of scalar operators, are explicitly summed using Bethe–Salpeter-type techniques. The structure constants and scaling dimensions extracted from OPE match exactly the eigenvalues and pole structure of the graph-building operators (Gromov et al., 2018). These are packaged elegantly via conformal block expansions and often involve multiple-zeta values at higher loops (Gromov et al., 2017, Kazakov et al., 2018).
Quantum Spectral Curve and Baxter Equation: The spectrum of anomalous dimensions is computed via a quantum spectral curve formalism. Here, a Baxter T–Q equation, derived from integrability and fixed by quantization conditions, governs the spectrum of single-trace operators. For example, for an operator $\mathcal{O}_J = \operatorname{tr}(\phi_1^J)$ , the scaling dimensions $\Delta = \Delta(\xi)$ solve a finite-difference Baxter equation with explicit all-loop quantization conditions (Gromov et al., 2017, Kazakov, 2018).
Thermodynamic Bethe Ansatz: The full spectrum of local multi-magnon operators is captured by TBA and Y-system equations, where the TBA free energy is often associated with the upper critical coupling of the model (Zamolodchikov's critical point), and dual TBA equations connect fishnet graphs to sigma models in $AdS_{D+1}$ (Basso et al., 2019).
IR and UV Structure: The model is conformal (except for special “short” operators); its amplitudes do not suffer from IR divergences, which enables direct LSZ reduction without necessitating IR regularization (Chicherin et al., 2022).

5. Dimensional Generalizations, Massive Deformations, and Extensions

Generalization to Higher/Lesser Dimensions: The structure of the fishnet theory is not limited to $D=4$ . Generalized models can be constructed for arbitrary $D$ by modifying the kinetic terms with fractional Laplacians. The integrability persists, and the dynamics is governed by an $SO(1,D+1)$ spin chain (Kazakov et al., 2018).
Massive Fishnets: Extensions to massive bi-scalar field theories are obtained via spontaneous symmetry breaking (VEVs for scalar fields) or by embedding the double-scaling limit into a Coulomb-branch deformed $\mathcal{N}=4$ SYM. Both constructions yield massive propagators but differ in whether masses enter as products or differences of VEVs. Integrability is preserved, and a massive version of the Yangian algebra is present; massive fishnet Feynman integrals exhibit this symmetry (the “massive Yangian”) (Loebbert et al., 2020).
Dynamical and Supersymmetric Fishnets: For more general chiral CFTs, the fishnet graph structure becomes “dynamical,” allowing multiple types of vertices and nontrivial mixing. Supersymmetric analogs (“superfishnet” and “super brick wall”) demonstrate integrability in superspace, with anomaly coefficients and OPE data accessible via superspace generalizations of the fishnet machinery (Kazakov et al., 2018, Kade, 23 Oct 2024, Kade, 3 Sep 2025).

6. Methods of Solution and Exact Results

Operator Algebra and SoV: Exact results for fishnet diagrams are achieved using separation of variables, Baxter operators, and star–triangle identities. In two dimensions, overlaps of separated-variable wavefunctions provide determinant representations for Basso–Dixon-type integrals, and the same strategy generalizes to higher-dimension and more complex graph topologies (Derkachov et al., 2018, Olivucci, 2023, Aprile et al., 2023).
All-Loop Results and Special Functions: Weak-coupling expansions yield coefficients involving polylogarithms (harmonic and elliptic) and multiple-zeta values. At strong coupling, four-point functions display classical exponential scaling, suggesting a semiclassical dual description (the “fishchain” or “bi-fishchain” model) and a connection to holographic dualities (Gromov et al., 2018, Huang, 2022).
Ward Identities and Bootstrap: Yangian Ward identities are derived for four-point ladder and Basso–Dixon integrals. These identities, phrased as inhomogeneous extensions of Appell hypergeometric equations, constrain the transcendental structure and may allow a bootstrap of the solutions (Corcoran et al., 2021).

7. Applications, Impact, and Generalizations

Analytic Control and Benchmarking: The bi-scalar fishnet theory stands out as one of the rare four-dimensional field theories where all-loop, multi-point, multi-loop integrals and OPE data can be explicitly computed and cross-checked—serving as an essential testbed for integrability-based methods and perturbative QFT in higher dimensions (Gromov et al., 2018, Chicherin et al., 2022).
Critical Coupling and Statistical Mechanics Links: The exact calculation of the model's critical coupling connects its non-perturbative behavior to classic results in statistical mechanics, such as the Zamolodchikov critical point for lattice models (Basso et al., 2019, Kade, 3 Sep 2025).
Extensions to Supersymmetric and Chiral Theories: Techniques initially developed for the bi-scalar fishnet theory—such as chain relations, star–triangle identities, and Yangian symmetry—generalize to a broad class of chiral, supersymmetric, and nonlocal CFTs, both in three and four dimensions, including nonunitary and logarithmic theories (Kade, 23 Oct 2024, Kade, 3 Sep 2025).
Future Directions: Continuing research seeks to exploit and extend fishnet integrability methods to theories with richer field content and to clarify connections to quantum integrable systems, higher spin/CFT dualities, and holography. The detailed analytic structure of multi-loop Feynman graphs in this context informs both QFT and statistical mechanics, potentially allowing the bootstrap of previously intractable observables.